OWA Operators in Decision Making∗

Robert Full er †

rfuller@ra.abo.fi ,http://www.abo.fi/ rfuller/robert.html

Abstract

In 1988 Ronald R. Yager [14] introduced a new aggregation techniquebased on the ordered weighted averaging (OWA) operators. The goal of thispaper is to present a short survey of of OWA operators and illustrate theirapplicability by a real-life example.

1 Fuzzy sets

Fuzzy sets were introduced by Zadeh (1965) as a means of representing and ma-nipulating data that was not precise, but rather fuzzy. Fuzzy logic provides aninference morphology that enables approximate human reasoning capabilities tobe applied to knowledge-based systems. The theory of fuzzy logic provides amathematical strength to capture the uncertainties associated with human cogni-tive processes, such as thinking and reasoning. The conventional approaches toknowledge representation lack the means for representating the meaning of fuzzyconcepts. As a consequence, the approaches based on first order logic and clas-sical probablity theory do not provide an appropriate conceptual framework fordealing with the representation of commonsense knowledge, since such knowl-edge is by its nature both lexically imprecise and noncategorical.

∗in: C. Carlsson ed., Exploring the Limits of Support Systems, TUCS General Publications,No. 3, Turku Centre for Computer Science, Abo, [ISBN 951-650-947-9, ISSN 1239-1905], 199685-104.†Presently a Donner Visiting Professor at Institute for Advanced Management Systems Re-

search, Abo Akademi University

1

The developement of fuzzy logic was motivated in large measure by the needfor a conceptual framework which can address the issue of uncertainty and lexicalimprecision.

Some of the essential characteristics of fuzzy logic relate to the following [25].

• In fuzzy logic, exact reasoning is viewed as a limiting case ofapproximate reasoning.

• In fuzzy logic, everything is a matter of degree.

• In fuzzy logic, knowledge is interpreted a collection of elasticor, equivalently, fuzzy constraint on a collection of variables.

• Inference is viewed as a process of propagation of elastic con-straints.

• Any logical system can be fuzzified.

There are two main characteristics of fuzzy systems that give them better per-formance for specific applications.

• Fuzzy systems are suitable for uncertain or approximate reasoning, espe-cially for the system with a mathematical model that is difficult to derive.

• Fuzzy logic allows decision making with estimated values under incompleteor uncertain information.

Definition 1.1 [23] LetX be a nonempty set. A fuzzy set A inX is characterizedby its membership function

µA : X → [0, 1]

and µA(x) is interpreted as the degree of membership of element x in fuzzy set Afor each x ∈ X .

It is clear thatA is completely determined by the set of tuples

A = {(x, µA(x))|x ∈ X}

Frequently we will write simplyA(x) instead ofµA(x). The family of allfuzzy (sub)sets inX is denoted byF(X). Fuzzy subsets of the real line are calledfuzzy quantities.

2

-2 -1 1 2 3

1

40

Figure 1: A discrete membership function for”x is close to 1”.

If X = {x1, . . . , xn} is afinite set andA is a fuzzy set inX then we often usethe notation

A = µ1/x1 + . . .+ µn/xn

where the termµi/xi, i = 1, . . . , n signifies thatµi is the grade of membership ofxi in A and the plus sign represents the union.

Example 1 Suppose we want to define the set of natural numbers ”close to 1”.This can be expressed by

A = 0.0/− 2 + 0.3/− 1 + 0.6/0 + 1.0/1 + 0.6/2 + 0.3/3 + 0.0/4.

Example 2 The membership function of the fuzzy set of real numbers ”close to1”, is can be defined as

A(t) = exp(−β(t− 1)2)

where β is a positive real number.

Example 3 Assume someone wants to buy a cheap car. Cheapcan be representedas a fuzzy set on a universe of prices, and depends on his purse. For instance, fromFig. 1.3. cheapis roughly interpreted as follows:

• Below 3000$ cars are considered as cheap, and prices make no real differ-ence to buyer’s eyes.

3

3000$ 6000$4500$

1

Figure 2: A membership function for”x is close to 1”.

Figure 3: Membership function of”cheap”.

• Between 3000$ and 4500$, a variation in the price induces a weak prefer-ence in favor of the cheapest car.

• Between 4500$ and 6000$, a small variation in the price induces a clearpreference in favor of the cheapest car.

• Beyond 6000$ the costs are too high (out of consideration).

Triangular norms were introduced by Schweizer and Sklar [12] to model thedistances in probabilistic metric spaces. In fuzzy sets theory triangular normsare extensively used to model logical connectiveand. Triangular conorms areextensively used to model logical connectiveor.

4

Definition 1.2 A mapping

T : [0, 1]× [0, 1]→ [0, 1]

is a triangular norm (t-norm for short) iff it is symmetric, associative, non-decreasingin each argument and T (a, 1) = a, for all a ∈ [0, 1].

Definition 1.3 A mapping

S : [0, 1]× [0, 1]→ [0, 1]

is a triangular co-norm (t-conorm for short) if it is symmetric, associative, non-decreasing in each argument and S(a, 0) = a, for all a ∈ [0, 1].

If T is a t-norm then the equality

S(a, b) := 1− T (1− a, 1− b)

defines a t-conorm and we say thatS is derived fromT . The basic t-norms andt-conorms pairs are

• minimum/maximum:

MIN(a, b) = min{a, b} = a ∧ b, MAX(a, b) = max{a, b} = a ∨ b

• Łukasiewicz:

LAND(a, b) = max{a+ b− 1, 0}, LOR(a, b) = min{a+ b, 1}

• probabilistic:PAND(a, b) = ab, POR(a, b) = a+ b− ab

• weak/strong:

WEAK(a, b) =

{min{a, b} if max{a, b} = 1

0 otherwise

STRONG(a, b) =

{max{a, b} if min{a, b} = 0

1 otherwise

5

• Hamacher:

HANDγ(a, b) =ab

γ + (1− γ)(a+ b− ab),

HORγ(a, b) =a+ b− (2− γ)ab

1− (1− γ)ab , γ ≥ 0

• Yager:

Y ANDp(a, b) = 1−min{1, p√

(1− a)p + (1− b)p},

Y ORp(a, b) = min{1, p√ap + bp}, p > 0

Definition 1.4 Let A and B be two fuzzy predicates defined on the real line R.Knowing that ’X is B’ is true, the degree of possibility that the proposition ’X isA’ is true, Π[A|B], is given by

Π[A|B] = sup{A(t) ∧B(t)|t ∈ R}, (1)

the degree of necessity that the proposition ’X is A’ is true, N [A|B], is given by

N [A|B] = 1− Π[¬A|B],

where A and B are the possibility distributions (for simplicity we write A insteadof µA) defined by the predicates A and B, respectively, and

(¬A)(t) = 1− A(t)

for any t. We can use any t-norm T in (1) to model the logical connective and:

Π[A|B] = sup{T (A(t), B(t))|t ∈ R}.

There are three important classes of fuzzy implication operators:

• S-implications: defined by

x→ y = S(n(x), y) (2)

where S is a t-conorm and n is a negation on [0, 1]. These implicationsarise from the Boolean formalism p → q = ¬p ∨ q. We shall use thefollowing S-implications: x → y = min{1 − x + y, 1} (Łukasiewitz) andx→ y = max{1− x, y} (Kleene-Dienes).

6

• R-implications: obtained by residuation of continuous t-norm T , i.e.

x→ y = sup{z ∈ [0, 1] | T (x, z) ≤ y}

These implications arise from the Intutionistic Logic formalism. We shalluse the following R-implication: x → y = 1 if x ≤ y and x → y = y ifx > y (Godel), x→ y = min{1− x+ y, 1} (Łukasiewitz)

• t-norm implications: if T is a t-norm then

x→ y = T (x, y)

Although these implications do not verify the properties of material impli-cation they are used as model of implication in many applications of fuzzylogic. We shall use the minimum-norm as t-norm implication (Mamdani).

Consider again the definition of t-norm-based possibility

Π[A|B] = sup{T (A(t), B(t))|t ∈ R}, (3)

where T is t-norm. Then for the measure of necessity of A, given B we get

N [A|B] = 1− Π[¬A|B] = 1− suptT (1− A(t), B(t))

Let S be a t-conorm derived from T , then

1− suptT (1− A(t), B(t)) = inf

t{1− T (1− A(t), B(t))} =

inft{S(1−B(t), A(t))} = inf

t{B(t)→ A(t)}

where the implication operator is defined in the sense of (2). That is,

N [A|B] = inft{B(t)→ A(t)}.

Let A andW be discrete fuzzy sets in the unit interval, such that

A = a1/(1/n) + a2/(2/n) + · · ·+ an/1

andW = w1/(1/n) + w2/(2/n) + · · ·+ wn/1

7

where n > 1, and the terms aj/(j/n) and wj/(j/n) signify that aj and wj are thegrades of membership of j/n in A andW , respectively, i.e.

A(j/n) = aj, W (j/n) = wj

for j = 1, . . . , n, and the plus sign represents the union. Then we get the followingsimple formula for the measure of necessity of A, givenW

N [A|W ] = minj=1,...,n

{W (j/n)→ A(j/n)} = minj=1,...,n

{wj → aj} (4)

and we use the notation

N [A|W ] = N [(a1, a2, . . . , an)|(w1, w2, . . . , wn)]

2 OWA Operators

In 1988 Ronald R. Yager [14] introduced a new aggregation technique based onthe ordered weighted averaging operators. OWA operators have been discussed ina large number of papers [7, 8, 9, 10, 15, 16, 17, 22].

Definition 2.1 An OWA operator of dimension n is a mapping F : Rn → R, thathas an associated n vector

w = (w1, w2, . . . , wn)T

such as wi ∈ [0, 1], 1 ≤ i ≤ n, and

n∑i=1

wi = w1 + · · ·+ wn = 1.

Furthermore

F (a1, . . . , an) =n∑j=1

wjbj = w1b1 + · · ·+ wnbn

where bj is the j-th largest element of the bag < a1, . . . , an >.

Example 4 Assume w = (0.4, 0.3, 0.2, 0.1)T then

F (0.7, 1, 0.2, 0.6) = 0.4× 1 + 0.3× 0.7 + 0.2× 0.6 + 0.1× 0.2 = 0.75.

8

A fundamental aspect of this operator is the re-ordering step, in particularan aggregate ai is not associated with a particular weight wi but rather a weightis associated with a particular ordered position of aggregate. When we view theOWA weights as a column vector we shall find it convenient to refer to the weightswith the low indices as weights at the top and those with the higher indices withweights at the bottom.

It is noted that different OWA operators are distinguished by their weightingfunction. We point out three important special cases of OWA aggregations:

• Max: In this case w∗ = (1, 0 . . . , 0)T and

Max(a1, . . . , an) = max{a1, . . . , an}.

• Min: In this case w∗ = (0, 0 . . . , 1)T and

Min(a1, . . . , an) = min{a1, . . . , an}.

• Average: In this case wA = (1/n, . . . , 1/n)T and

FA(a1, . . . , an) =a1 + · · ·+ an

n

We can see the OWA operators have the basic properties associated with an aver-aging operator (commutative, monotonic and idempotent).

A window type OWA operator takes the average of them arguments about thecenter. For this class of operators we have

wi =

0 if i < k1/m if k ≤ i < k +m0 if i ≥ k +m

For example, let m = 3 and k = 2. Then the weights of this window type OWAoperator are calculated as w1 = 0, w2 = w3 = w4 = 1/3, w5 = 0. This operatortakes the arithmetic mean of all but the best and the worst scores of an alternative.

Compensative connectives have the property that a higher degree of satisfac-tion of one of the criteria can compensate for a lower degree of satisfaction ofanother criterion. Oring the criteria means full compensation and anding the cri-teria means no compensation. In order to classify OWA operators in regard to

9

k k+m-11 n

1/m

Figure 4: Window type OWA operator.

their location between and and or, Yager [14] introduced a measure of orness,associated with any vector w as follows

orness(w) =1

n− 1

n∑i=1

(n− i)wi

It is easy to see that for any w the orness(w) is always in the unit interval. Fur-thermore, note that the nearer w is to an or, the closer its measure is to one; whilethe nearer it is to an and, the closer is to zero. Generally, an OWA operator withmuch of nonzero weights near the top will be an orlike operator,

orness(w) ≥ 0.5

and when much of the weights are nonzero near the bottom, the OWA operatorwill be andlike

andness(w) := 1− orness(w) ≥ 0.5.

The following theorem shows that as we move weight up the vector we increasethe orness, while moving weight down causes us to decrease orness(W ).

Theorem 2.1 [16] Assume W and W ′ are two n-dimensional OWA vectors suchthat

W = (w1, . . . , wn)T , W ′ = (w1, . . . , wj + ε, . . . , wk − ε, . . . , wn)T

where ε > 0, j < k. Then orness(W ′) > orness(W ).

Example 5 Let w = (0.8, 0.2, 0.0)T . Then

orness(w) =1

3(2× 0.8 + 0.2) = 0.6

10

andandness(w) = 1− orness(w) = 1− 0.6 = 0.4.

This means that the OWA operator, defined by

F (a1, a2, a3) = 0.8b1 + 0.2b2 + 0.0b3 = 0.8b1 + 0.2b2

where bj is the j-th largest element of the bag < a1, a2, a3 >, is an orlike aggre-gation.

In [14] Yager defined the measure of dispersion (or entropy) of an OWA vectorby

disp(w) = −∑i

wi lnwi.

We can see when using the OWA operator as an averaging operator Disp(W )measures the degree to which we use all the aggregates equally.

If F is an OWA aggregation with weights wi the dual of F denoted F , is anOWA aggregation of the same dimention where with weights wi

wi = wn−i+1.

We can easily see that if F and F are duals then

disp(F ) = disp(F )

orness(F ) = 1− orness(F ) = andness(F )

Thus is F is orlike its dual is andlike.

Example 6 Let w = (0.3, 0.2, 0.1, 0.4)T . Then

w = (0.4, 0.1, 0.2, 0.3)T .

and

orness(F ) =1

3(3× 0.3 + 2× 0.2 + 0.1) ≈ 0.466,

orness(F ) =≈ 0.533.

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An important application of the OWA operators is in the area of quantifierguided aggregations [14]. Assume

{A1, . . . , An}

is a collection of criteria. Let x be an object such that for any criterionAi,Ai(x) ∈[0, 1] indicates the degree to which this criterion is satisfied by x. If we want tofind out the degree to which x satisfies ”all the criteria” denoting this by D(x),we get following Bellman and Zadeh [1].

D(x) = min{A1(x), . . . , An(x)}

In this case we are essentially requiring x to satisfy A1 and A2 and . . . and An.

If we desire to find out the degree to which x satisfies ”at least one of thecriteria” , denoting this E(x), we get

E(x) = max{A1(x), . . . , An(x)}

In this case we are requiring x to satisfy A1 or A2 or . . . or An.

In many applications rather than desiring that a solution satisfies one of theseextreme situations, ”all” or ”at least one” , we may require that x satisfies mostor at least half of the criteria. Drawing upon Zadeh’s concept [24] of linguisticquantifiers we can accomplish these kinds of quantifier guided aggregations.

Definition 2.2 A quantifier Q is called

• regular monotonically non-decreasing if

Q(0) = 0, Q(1) = 1, if r1 > r2 then Q(r1) ≥ Q(r2).

• regular monotonically non-increasing if

Q(0) = 1, Q(1) = 0, if r1 < r2 then Q(r1) ≥ Q(r2).

• regular unimodal if

Q(0) = Q(1) = 0, Q(r) = 1 for a ≤ r ≤ b,

r2 ≤ r1 ≤ a then Q(r1) ≥ Q(r2), r2 ≥ r1 ≥ b then Q(r2) ≤ Q(r1).

12

Figure 5: Monotone linguistic quantifiers.

Figure 6: Unimodal linguistic quantifier.

13

1/n 2/n 3/n

w1

w2

w3

With ai = Ai(x) the overall valuation of x is FQ(a1, . . . , an) where FQ is anOWA operator. The weights associated with this quantified guided aggregationare obtained as follows

wi = Q(i

n)−Q(

i− 1

n), i = 1, . . . , n. (5)

Fig. 7 graphically shows the operation involved in determining the OWA weightsdirectly from the quantifier guiding the aggregation.

Figure 7: Determining weights from a quantifier.

Let us look at the weights generated from some basic types of quantifiers. Thequantifier, for all Q∗, is defined such that

Q∗(r) =

{0 for r < 1,1 for r = 1.

Using our method for generating weights

wi = Q∗(i

n)−Q∗(

i− 1

n)

we get

wi =

{0 for i < n,1 for i = n.

14

1

1

1

1

Figure 8: The quantifier all.

Figure 9: The quantifier there exists.

This is exactly what we previously denoted asW∗.

For the quantifier there exists we have

Q∗(r) =

{0 for r = 0,1 for r > 0.

In this case we getw1 = 1, wi = 0, for i �= 1.

This is exactly what we denoted asW ∗.

Consider next the quantifier defined by

Q(r) = r.

15

1

1

Figure 10: The identity quantifier.

This is an identity or linear type quantifier.

In this case we get

wi = Q(i

n)−Q(

i− 1

n) =

i

n−i− 1

n=

1

n.

This gives us the pure averaging OWA aggregation operator.

The standard degree of orness associated with a Regular Increasing Monotone(RIM) linguistic quantifier Q

orness(Q) =

∫ 1

0

Q(r) dr

is equal to the area under the quantifier [20]. This definition for the measure oforness of quantifier provides a simple useful method for obtaining this measure.

Consider the family of RIM quantifiers

Qα(r) = rα, α ≥ 0. (6)

It is clear that

orness(Qα) =

∫ 1

0

rα dr =1

α+ 1

and orness(Qα) < 0.5 for α > 1, orness(Qα) = 0.5 for α = 1 and orness(Qα) >0.5 for α < 1.

For example, if α = 2 then we get

orness(Qα) =

∫ 1

0

r2 dr =1

2 + 1=

1

3

16

Figure 11: Risk averse and risk pro RIM linguistic quanitfiers.

3 Case study

We illustrate the applicability of OWA operators by a doctoral student selectionproblem at the Graduate School of Turku Centre for Computer Science (see [4]for details).

The problem of selecting young promising doctoral researchers can be seento consist of three components. The first component is a collection

X = {x1, . . . , xp}

of applicants for the Ph.D. program. The second component is a collection of 6criteria (see Table 1) which are considered relevant in the ranking process.

17

Research interests (excellent) (average) (weak)

- Fit in research groups © © ©

- On the frontier of research © © ©

- Contributions © © ©

Academic background

- University © © ©

- Grade average © © ©

- Time for acquiring degree © © ©

Letters of recommendation Y N

Knowledge of English Y N

Table 1 Evaluation sheet.

For simplicity we suppose that all applicants are young and have Master’sdegree acquired more than one year before. In this case all the criteria are mean-ingful, and are of approximately the same importance.

The third component is a group of 11 experts whose opinions are solicited inranking the alternatives. The experts are selected from the following 9 researchgroups

So we have a Multi Expert-Multi Criteria Decision Making (ME-MCDM)problem. The ranking system described in the following is a two stage process.In the first stage, individual experts are asked to provide an evaluation of the al-ternatives. This evaluation consists of a rating for each alternative on each ofthe criteria, where the ratings are chosen from the scale {1, 2, 3}, where 3 standsfor excellent, 2 stands for average and 1 means weak performance. Each expertprovides a 6-tuple

(a1, . . . , a6)

18

for each applicant, where ai ∈ {1, 2, 3}, i = 1, . . . , 6. The next step in the processis to find the overall evaluation for an alternative by a given expert using an OWAoperator derived from an appropriate linguistic quantifier from family (6).

We search for an index α ≥ 0 such that the associated linguistic quantifier Qα

from the family (6) approximates the experts’ preferences as much as possible.After interviewing the experts we found that all of them agreed on the followingprinciples

(i) if an applicant has more than two weak performances then his overall per-formance should be less than two,

(ii) if an applicant has maximum two weak performances then his overall per-formance should be more than two,

(iii) if an applicant has all but one excellent performances then his overall per-formance should be about 2.75,

(iv) if an applicant has three weak performances and one of them is on the crite-rion on the frontier of research then his overall performance should not beabove 1.5,

From (i) and (ii) we get1 < α ≤ 1.293,

which means that Qα should be andlike (or risk averse) quantifier with a degreeof compensation just below the arithmetic average.

It is easy to verify that (iii) and (iv) can not be satisfied by any quantifierQα, 1 < α ≤ 1.293, from the family (6). In fact, (iii) requires that α ≈ 0.732which is smaller than 1 and (iv) can be satisfied if α ≥ 2 which is bigger than1.293. Rules (iii) and (iv) have priority whenever they are applicable.

In the second stage the technique for combining the expert’s evaluation toobtain an overall evaluation for each alternative is based upon the OWA operators.Each applicant is represented by an 11-tuple

(b1, . . . , b11)

where bi ∈ [1, 3] is the unit score derived from the i-th expert’s ratings. Wesuppose that the bi’s are organized in descending order, i.e. bi can be seen as theworst of the i-th top scores.

19

Taking into consideration that the experts are selected from 9 different researchgroups there exists no applicant that scores overall well on the first criterion ”Fit inresearch group” . After a series of negotiations all experts agreed that the supportof at least four experts is needed for qualification of the applicant.

Since we have 11 experts, applicants are evaluated based on their top fourscores

(b1, . . . , b4)

and if at least three experts agree that the applicant is excellent then his final scoreshould be 2.75 which is a cut-off value for the best student. That is

Fα(3, 3, 3, 1) = 3× (w1 + w2 + w3) + w4 = 2.75,

that is,

3×[3

4

]α+ 1−

[3

4

]α= 2.75 ⇐⇒

[3

4

]α= 0.875 ⇐⇒ α ≈ 0.464

So in the second stage we should choose an orlike OWA operator with α ≈ 0.464for aggregating the top six scores of the applicant to find the final score.

If the final score is less than 2 then the applicant is disqualified and if the finalscore is at least 2.5 then the scholarship should be awarded to him. If the finalscore is between 2 and 2.5 then the scholarship can be awarded to the applicantpending on the total number of scholarships available.

Example 7 Let us choose α = 1.2 for the aggregation of the ratings in the firststage. Consider some applicant with the following scores

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Criteria C1 C2 C3 C4 C5 C6

Expert 1 3 2 3 2 3 1Expert 2 2 3 3 2 3 2Expert 3 2 2 3 2 2 1Expert 4 3 2 3 3 3 2Expert 5 2 2 3 2 3 1Expert 6 3 2 3 2 3 1Expert 7 1 2 3 2 3 2Expert 8 1 2 3 2 3 1Expert 9 1 2 2 2 3 2Expert 10 1 2 2 3 3 1Expert 11 1 2 2 2 2 1

The weights associated with this linguistic quantifier are

(0.116, 0.151, 0.168, 0.180, 0.189, 0.196)

After re-ordering the scores in descending order we get the following table

Unit score

Expert 1 3 3 3 2 2 1 2.239Expert 2 3 3 3 2 2 2 2.435Expert 3 3 2 2 2 2 1 1.920Expert 4 3 3 3 3 2 2 2.615Expert 5 3 3 2 2 2 1 2.071Expert 6 3 3 3 2 2 1 2.239Expert 7 3 3 2 2 2 1 2.071Expert 8 3 3 2 2 1 1 1.882Expert 9 3 2 2 2 2 1 1.920Expert 10 3 3 2 2 1 1 1.882Expert 11 2 2 2 2 1 1 1.615

In the second stage we choose α = 0.464 and obtain the following weights

(0.526, 0.199, 0.150, 0.125).

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The best four scores of the applicant are

(2.615, 2.435, 2.239, 2.239).

The final score is computed as

Fα(2.615, 2.435, 2.239, 2.239) = 2.475.

So the applicant has good chances to get the scholarship.

4 Summary

In a decision process the idea of trade-offs corresponds to viewing the globalevaluation of an action as lying between the worst and the best local ratings. Thisoccurs in the presence of conflicting goals, when a compensation between thecorresponding compabilities is allowed. OWA operators can realize trade-offsbetween objectives, by allowing a positive compensation between ratings, i.e. ahigher degree of satisfaction of one of the criteria can compensate for a lower de-gree of satisfaction of another criteria to a certain extent. OWA operators providefor any level of compensation lying between the logical and and or. If we aregiven a decision problem then we find an appropriate OWA aggregation operatorfrom some rules and/or samples determined by the decision makers.

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